arXiv subjects

Karl-Mikael Perfekt

Publications and source records attributed to Karl-Mikael Perfekt.

At least 37 records · Page 2Linked to original sources

arXiv2021

Orthogonal decomposition of composition operators on the $H^2$ space of Dirichlet series

Let $\mathscr{H}^2$ denote the Hilbert space of Dirichlet series with square-summable coefficients. We study composition operators $\mathscr{C}_φ$ on $\mathscr{H}^2$ which are generated by symbols of the form $φ(s) = c_0s + \sum_{n\geq1} c_n n^{-s}$, in the case that $c_0 \geq 1$. If only a subset $\mathbb{P}$ of prime numbers features in the Dirichlet series of $φ$, then the operator $\mathscr{C}_φ$ admits an associated orthogonal decomposition. Under sparseness assumptions on $\mathbb{P}$ we use this to asymptotically estimate the approximation numbers of $\mathscr{C}_φ$. Furthermore, in the case that $φ$ is supported on a single prime number, we affirmatively settle the problem of describing the compactness of $\mathscr{C}_φ$ in terms of the ordinary Nevanlinna counting function. We give detailed applications of our results to affine symbols and to angle maps.

math.FA↗

arXiv2021

The Complex-Scaled Half-Space Matching Method

The Half-Space Matching (HSM) method has recently been developed as a new method for the solution of 2D scattering problems with complex backgrounds, providing an alternative to Perfectly Matched Layers (PML) or other artificial boundary conditions. Based on half-plane representations for the solution, the scattering problem is rewritten as a system of integral equations in which the unknowns are restrictions of the solution to the boundaries of a finite number of overlapping half-planes contained in the domain: this integral equation system is coupled to a standard finite element discretisation localised around the scatterer. While satisfactory numerical results have been obtained for real wavenumbers, wellposedness and equivalence to the original scattering problem have been established only for complex wavenumbers. In the present paper, by combining the HSM framework with a complex-scaling technique, we provide a new formulation for real wavenumbers which is provably well-posed and has the attraction for computation that the complex-scaled solutions of the integral equation system decay exponentially at infinity. The analysis requires the study of double-layer potential integral operators on intersecting infinite lines, and their analytic continuations. The effectiveness of the method is validated by preliminary numerical results.

math.NA↗

arXiv2021

Infinitely many embedded eigenvalues for the Neumann-Poincaré operator in 3D

This article constructs a surface whose Neumann-Poincaré (NP) integral operator has infinitely many eigenvalues embedded in its essential spectrum. The surface is a sphere perturbed by smoothly attaching a conical singularity, which imparts essential spectrum. Rotational symmetry allows a decomposition of the operator into Fourier components. Eigenvalues of infinitely many Fourier components are constructed so that they lie within the essential spectrum of other Fourier components and thus within the essential spectrum of the full NP operator. The proof requires the perturbation to be sufficiently small, with controlled curvature, and the conical singularity to be sufficiently flat.

math.FA↗

arXiv2021

A mean counting function for Dirichlet series and compact composition operators

We introduce a mean counting function for Dirichlet series, which plays the same role in the function theory of Hardy spaces of Dirichlet series as the Nevanlinna counting function does in the classical theory. The existence of the mean counting function is related to Jessen and Tornehave's resolution of the Lagrange mean motion problem. We use the mean counting function to describe all compact composition operators with Dirichlet series symbols on the Hardy--Hilbert space of Dirichlet series, thus resolving a problem which has been open since the bounded composition operators were described by Gordon and Hedenmalm. The main result is that such a composition operator is compact if and only if the mean counting function of its symbol satisfies a decay condition at the boundary of a half-plane.

math.FA↗

arXiv2020

Plasmonic eigenvalue problem for corners: limiting absorption principle and absolute continuity in the essential spectrum

We consider the plasmonic eigenvalue problem for a general 2D domain with a curvilinear corner, studying the spectral theory of the Neumann--Poincaré operator of the boundary. A limiting absorption principle is proved, valid when the spectral parameter approaches the essential spectrum. Putting the principle into use, it is proved that the corner produces absolutely continuous spectrum of multiplicity 1. The embedded eigenvalues are discrete. In particular, there is no singular continuous spectrum.

math.SP↗

arXiv2020

Rectangular summation of multiple Fourier series and multi-parametric capacity

We consider the class of multiple Fourier series associated with functions in the Dirichlet space of the polydisc. We prove that every such series is summable with respect to unrestricted rectangular partial sums, everywhere except for a set of zero multi-parametric logarithmic capacity. Conversely, given a compact set in the torus of zero capacity, we construct a Fourier series in the class which diverges on this set, in the sense of Pringsheim. We also prove that the multi-parametric logarithmic capacity characterizes the exceptional sets for the radial variation and radial limits of Dirichlet space functions. As a by-product of the methods of proof, the results also hold in the vector-valued setting.

math.CA↗

arXiv2019

Norms of composition operators on the $H^2$ space of Dirichlet series

We consider composition operators $\mathscr{C}_φ$ on the Hardy space of Dirichlet series $\mathscr{H}^2$, generated by Dirichlet series symbols $φ$. We prove two different subordination principles for such operators. One concerns affine symbols only, and is based on an arithmetical condition on the coefficients of $φ$. The other concerns general symbols, and is based on a geometrical condition on the boundary values of $φ$. Both principles are strict, in the sense that they characterize the composition operators of maximal norm generated by symbols having given mapping properties. In particular, we generalize a result of J. H. Shapiro on the norm of composition operators on the classical Hardy space of the unit disc. Based on our techniques, we also improve the recently established upper and lower norm bounds in the special case that $φ(s) = c + r2^{-s}$. A number of other examples are given.

math.FA↗

arXiv2019

Atomic decompositions, two stars theorems, and distances for the Bourgain-Brezis-Mironescu space and other big spaces

Given a Banach space $E$ with a supremum-type norm induced by a collection of operators, we prove that $E$ is a dual space and provide an atomic decomposition of its predual. We apply this result, and some results obtained previously by one of the authors, to the function space $\mathcal{B}$ introduced recently by Bourgain, Brezis, and Mironescu. This yields an atomic decomposition of the predual $\mathcal{B}_\ast$, the biduality result that $\mathcal{B}_0^\ast = \mathcal{B}_\ast$ and $\mathcal{B}_\ast^\ast = \mathcal{B}$, and a formula for the distance from an element $f \in \mathcal{B}$ to $\mathcal{B}_0$.

math.FA↗

arXiv2018

Some properties related to trace inequalities for the multi-parameter Hardy operators on poly-trees

In this note we investigate the multi-parameter Potential Theory on the weighted $d$-tree (Cartesian product of several copies of uniform dyadic tree), which is connected to the discrete models of weighted Dirichlet spaces on the polydisc. We establish some basic properties of the respective potentials, capacities and equilibrium measures (in particular in the case of product polynomial weights). We explore multi-parameter Hardy inequality and its trace measures, and discuss some open problems of potential-theoretic and combinatorial nature.

math.CV↗

arXiv2018

The transmission problem on a three-dimensional wedge

We consider the transmission problem for the Laplace equation on an infinite three-dimensional wedge, determining the complex parameters for which the problem is well-posed, and characterizing the infinite multiplicity nature of the spectrum. This is carried out in two formulations leading to rather different spectral pictures. One formulation is in terms of square integrable boundary data, the other is in terms of finite energy solutions. We use the layer potential method, which requires the harmonic analysis of a non-commutative non-unimodular group associated with the wedge.

math.AP↗

arXiv2017

On the spaces of bounded and compact multiplicative Hankel operators

A multiplicative Hankel operator is an operator with matrix representation $M(α) = \{α(nm)\}_{n,m=1}^\infty$, where $α$ is the generating sequence of $M(α)$. Let $\mathcal{M}$ and $\mathcal{M}_0$ denote the spaces of bounded and compact multiplicative Hankel operators, respectively. In this note it is shown that the distance from an operator $M(α) \in \mathcal{M}$ to the compact operators is minimized by a nonunique compact multiplicative Hankel operator $N(β) \in \mathcal{M}_0$, $$\|M(α) - N(β)\|_{\mathcal{B}(\ell^2(\mathbb{N}))} = \inf \left \{\|M(α) - K \|_{\mathcal{B}(\ell^2(\mathbb{N}))} \, : \, K \colon \ell^2(\mathbb{N}) \to \ell^2(\mathbb{N}) \textrm{ compact} \right\}.$$ Intimately connected with this result, it is then proven that the bidual of $\mathcal{M}_0$ is isometrically isomorphic to $\mathcal{M}$, $\mathcal{M}_0^{\ast \ast} \simeq \mathcal{M}$. It follows that $\mathcal{M}_0$ is an M-ideal in $\mathcal{M}$. The dual space $\mathcal{M}_0^\ast$ is isometrically isomorphic to a projective tensor product with respect to Dirichlet convolution. The stated results are also valid for small Hankel operators on the Hardy space $H^2(\mathbb{D}^d)$ of a finite polydisk.

math.FA↗

arXiv2017

Nehari's theorem for convex domain Hankel and Toeplitz operators in several variables

We prove Nehari's theorem for integral Hankel and Toeplitz operators on simple convex polytopes in several variables. A special case of the theorem, generalizing the boundedness criterion of the Hankel and Toeplitz operators on the Paley-Wiener space, reads as follows. Let $Ξ= (0,1)^d$ be a $d$-dimensional cube, and for a distribution $f$ on $2Ξ$, consider the Hankel operator $$Γ_f (g)(x)=\int_Ξ f(x+y) g(y) \, dy, \quad x \inΞ.$$ Then $Γ_f$ extends to a bounded operator on $L^2(Ξ)$ if and only if there is a bounded function $b$ on $\mathbb{R}^d$ whose Fourier transform coincides with $f$ on $2Ξ$. This special case has an immediate application in matrix extension theory: every finite multi-level block Toeplitz matrix can be boundedly extended to an infinite multi-level block Toeplitz matrix. In particular, block Toeplitz operators with blocks which are themselves Toeplitz, can be extended to bounded infinite block Toeplitz operators with Toeplitz blocks.

math.FA↗

arXiv2017

The spectra of harmonic layer potential operators on domains with rotationally symmetric conical points

We study the adjoint of the double layer potential associated with the Laplacian (the adjoint of the Neumann-Poincaré operator), as a map on the boundary surface $Γ$ of a domain in $\mathbb{R}^3$ with conical points. The spectrum of this operator directly reflects the well-posedness of related transmission problems across $Γ$. In particular, if the domain is understood as an inclusion with complex permittivity $ε$, embedded in a background medium with unit permittivity, then the polarizability tensor of the domain is well-defined when $(ε+1)/(ε-1)$ belongs to the resolvent set in energy norm. We study surfaces $Γ$ that have a finite number of conical points featuring rotational symmetry. On the energy space, we show that the essential spectrum consists of an interval. On $L^2(Γ)$, i.e. for square-integrable boundary data, we show that the essential spectrum consists of a countable union of curves, outside of which the Fredholm index can be computed as a winding number with respect to the essential spectrum. We provide explicit formulas, depending on the opening angles of the conical points. We reinforce our study with very precise numerical experiments, computing the energy space spectrum and the spectral measures of the polarizability tensor in two different examples. Our results indicate that the densities of the spectral measures may approach zero extremely rapidly in the continuous part of the energy space spectrum.

math.AP↗

arXiv2017

On the spectrum of the multiplicative Hilbert matrix

We study the multiplicative Hilbert matrix, i.e. the infinite matrix with entries $(\sqrt{mn}\log(mn))^{-1}$ for $m,n\geq2$. This matrix was recently introduced within the context of the theory of Dirichlet series, and it was shown that the multiplicative Hilbert matrix has no eigenvalues and that its continuous spectrum coincides with $[0,π]$. Here we prove that the multiplicative Hilbert matrix has no singular continuous spectrum and that its absolutely continuous spectrum has multiplicity one. Our argument relies on the tools of spectral perturbation theory and scattering theory. Finding an explicit diagonalisation of the multiplicative Hilbert matrix remains an interesting open problem.

math.FA↗

arXiv2017

On Helson matrices: moment problems, non-negativity, boundedness, and finite rank

We study Helson matrices (also known as multiplicative Hankel matrices), i.e. infinite matrices of the form $M(α) = \{α(nm)\}_{n,m=1}^\infty$, where $α$ is a sequence of complex numbers. Helson matrices are considered as linear operators on $\ell^2(\mathbb{N})$. By interpreting Helson matrices as Hankel matrices in countably many variables we use the theory of multivariate moment problems to show that $M(α)$ is non-negative if and only if $α$ is the moment sequence of a measure $μ$ on $\mathbb{R}^\infty$, assuming that $α$ does not grow too fast. We then characterize the non-negative bounded Helson matrices $M(α)$ as those where the corresponding moment measures $μ$ are Carleson measures for the Hardy space of countably many variables. Finally, we give a complete description of the Helson matrices of finite rank, in parallel with the classical Kronecker theorem on Hankel matrices.

math.FA↗

arXiv2017

Contractive inequalities for Bergman spaces and multiplicative Hankel forms

We consider sharp inequalities for Bergman spaces of the unit disc, establishing analogues of the inequality in Carleman's proof of the isoperimetric inequality and of Weissler's inequality for dilations. By contractivity and a standard tensorization procedure, the unit disc inequalities yield corresponding inequalities for the Bergman spaces of Dirichlet series. We use these results to study weighted multiplicative Hankel forms associated with the Bergman spaces of Dirichlet series, reproducing most of the known results on multiplicative Hankel forms associated with the Hardy spaces of Dirichlet series. In addition, we find a direct relationship between the two type of forms which does not exist in lower dimensions. Finally, we produce some counter-examples concerning Carleson measures on the infinite polydisc.

math.FA↗

arXiv2017

Endpoint compactness of singular integrals and perturbations of the Cauchy Integral

We prove sufficient and necessary conditions for compactness of Calderón-Zygmund operators on the endpoint from $L^{\infty }(\mathbb R)$ into ${\rm CMO}(\mathbb R)$. We use this result to prove compactness on $L^{p}(\mathbb R)$ with $1<p<\infty $ of certain perturbations of the Cauchy integral on curves with normal derivatives satisfying a ${\rm CMO}$-condition.

math.CA↗

arXiv2016

Volterra operators on Hardy spaces of Dirichlet series

For a Dirichlet series symbol $g(s) = \sum_{n \geq 1} b_n n^{-s}$, the associated Volterra operator $\mathbf{T}_g$ acting on a Dirichlet series $f(s)=\sum_{n\ge 1} a_n n^{-s}$ is defined by the integral $f\mapsto -\int_{s}^{+\infty} f(w)g'(w)\,dw$. We show that $\mathbf{T}_g$ is a bounded operator on the Hardy space $\mathcal{H}^p$ of Dirichlet series with $0 < p < \infty$ if and only if the symbol $g$ satisfies a Carleson measure condition. When appropriately restricted to one complex variable, our condition coincides with the standard Carleson measure characterization of ${\operatorname{BMOA}}(\mathbb{D})$. A further analogy with classical ${\operatorname{BMO}}$ is that $\exp(c|g|)$ is integrable (on the infinite polytorus) for some $c > 0$ whenever $\mathbf{T}_g$ is bounded. In particular, such $g$ belong to $\mathcal{H}^p$ for every $p < \infty$. We relate the boundedness of $\mathbf{T}_g$ to several other ${\operatorname{BMO}}$ type spaces: ${\operatorname{BMOA}}$ in half-planes, the dual of $\mathcal{H}^1$, and the space of symbols of bounded Hankel forms. Moreover, we study symbols whose coefficients enjoy a multiplicative structure and obtain coefficient estimates for $m$-homogeneous symbols as well as for general symbols. Finally, we consider the action of $\mathbf{T}_g$ on reproducing kernels for appropriate sequences of subspaces of $\mathcal{H}^2$. Our proofs employ function and operator theoretic techniques in one and several variables; a variety of number theoretic arguments are used throughout the paper in our study of special classes of symbols $g$.

math.FA↗